\section{1.13} 
\begin{frame}[allowframebreaks]{1.13. }

\vspace{-0.4cm}

1.13. Holonomic complexes. 

A complex $M^\bullet \in D^b(\mathcal{D}_X)$ is said to be holonomic if the derived complex $H^iM^\bullet$ is holonomic. 

We let $D^b_{\mathrm{hol}}(\mathcal{D}_X)$ be the category of bounded holonomic complexes.

Note it is not required that the $M^k$ be holonomic. 

It would be more precise to call $M^\bullet$ a complex with holonomic cohomology, as in V. 

We shall follow the tradition by speaking of holonomic complexes. 

Of course, it is the only sensible notion if we put ourselves squarely in the derived category, as we are supposed to do, since if $M^\bullet$ and $N^\bullet$ are h.i. complexes and one of them consists of holonomic $\mathcal{D}$-modules, the other one need not be so.

It follows from 1.12 that if $M^\bullet$ consists of holonomic $\mathcal{D}$-modules, then it is a holonomic complex. 

There is therefore a natural map
\[
D^b(\mathrm{hol}(\mathcal{D}_X)) \to D^b_h(\mathcal{D}_X).
\]

Beilinson has recently proved that it is an equivalence of categories [B1], but we shall not need this fact.

\end{frame}

